Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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ANALYSIS OF A DELAY PREY-PREDATOR MODEL WITH DISEASE IN THE PREY SPECIES ONLY

  • Zhou, Xueyong (COLLEGE OF MATHEMATICS AND INFORMATION SCIENCE XINYANG NORMAL UNIVERSITY) ;
  • Shi, Xiangyun (COLLEGE OF MATHEMATICS AND INFORMATION SCIENCE XINYANG NORMAL UNIVERSITY) ;
  • Song, Xinyu (COLLEGE OF MATHEMATICS AND INFORMATION SCIENCE XINYANG NORMAL UNIVERSITY)
  • Published : 2009.07.01
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Abstract

In this paper, a three-dimensional eco-epidemiological model with delay is considered. The stability of the two equilibria, the existence of Hopf bifurcation and the permanence are investigated. It is found that Hopf bifurcation occurs when the delay ${\tau}$ passes though a sequence of critical values. The estimation of the length of delay to preserve stability has also been calculated. Numerical simulation with a hypothetical set of data has been done to support the analytical findings.

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References

  1. M. A. Aziz-Alaoui, Study of a Leslie-Gower-type tritrophic population model, Chaos Solitons Fractals 14 (2002), no. 8, 1275–1293 https://doi.org/10.1016/S0960-0779(02)00079-6
  2. M. A. Aziz-Alaoui and M. Daher Okiye, Boundedness and global stability for a predatorprey model with modified Leslie-Gower and Holling-type II schemes, Appl. Math. Lett. 16 (2003), no. 7, 1069–1075 https://doi.org/10.1016/S0893-9659(03)90096-6
  3. E. Beretta and Y. Takeuchi, Convergence results in SIR epidemic models with varying population sizes, Nonlinear Anal. 28 (1997), no. 12, 1909–1921 https://doi.org/10.1016/S0362-546X(96)00035-1
  4. G. Butler, H. I. Freedman, and P. Waltman, Uniformly persistent systems, Proc. Amer. Math. Soc. 96 (1986), no. 3, 425–430 https://doi.org/10.1090/S0002-9939-1986-0822433-4
  5. J. Chattopadhyay and N. Bairagi, Pelicans at risk in Salton Sea-an eco-epidemiological model, Ecolog. Modell. 136 (2001), 103–112 https://doi.org/10.1016/S0304-3800(00)00350-1
  6. J. Chattopadhyay, P. D. N. Srinivasu, and N. Bairagi, Pelican at risk in Salton Sea-an ecoepidemiological model-II, Ecolog. Modell. 167 (2003) 199–211 https://doi.org/10.1016/S0304-3800(03)00187-X
  7. K. L. Cooke and P. van den Driessche, On zeroes of some transcendental equations, Funkcial. Ekvac. 29 (1986), no. 1, 77–90
  8. H. I. Freedman and P. Moson, Persistence definitions and their connections, Proc. Amer. Math. Soc. 109 (1990), no. 4, 1025–1033 https://doi.org/10.1090/S0002-9939-1990-1012928-6
  9. H. Freedman and V. S. H. Rao, The trade-off between mutual interference and time lags in predator-prey systems, Bull. Math. Biol. 45 (1983), no. 6, 991–1004 https://doi.org/10.1007/BF02458826
  10. H. I. Freedman and P.Waltman, Persistence in a model of three competitive populations, Math. Biosci. 73 (1985), no. 1, 89–101 https://doi.org/10.1016/0025-5564(85)90078-1
  11. J. K. Hale and P. Waltman, Persistence in infinite-dimensional systems, SIAM J. Math. Anal. 20 (1989), no. 2, 388–395 https://doi.org/10.1137/0520025
  12. H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev. 42 (2000), no. 4, 599–653 https://doi.org/10.1137/S0036144500371907
  13. H. W. Hethcote, W. Wang, L. Han, and Z. Ma, A predator-prey model with infected prey, Theor. Pop. Biol. 66 (2004), 259–268 https://doi.org/10.1016/j.tpb.2004.06.010
  14. S. B. Hsu and T. W. Hwang, Hopf bifurcation analysis for a predator-prey system of Holling and Leslie type, Taiwanese J. Math. 3 (1999), no. 1, 35–53 https://doi.org/10.11650/twjm/1500407053
  15. S. B. Hsu, T. W. Hwang, and Y. Kuang, Global analysis of the Michaelis-Menten-type ratio-dependent predator-prey system, J. Math. Biol. 42 (2001), no. 6, 489–506 https://doi.org/10.1007/s002850100079
  16. T. W. Hwang, Uniqueness of the limit cycle for Gause-type predator-prey systems, J. Math. Anal. Appl. 238 (1999), no. 1, 179–195 https://doi.org/10.1006/jmaa.1999.6520
  17. Y. Kuang and E. Beretta, Global qualitative analysis of a ratio-dependent predator-prey system, J. Math. Biol. 36 (1998), no. 4, 389–406 https://doi.org/10.1007/s002850050105
  18. W. M. Liu, H. W. Hethcote, and S. A. Levin, Dynamical behavior of epidemiological models with nonlinear incidence rates, J. Math. Biol. 25 (1987), no. 4, 359–380 https://doi.org/10.1007/BF00277162
  19. W. Ma, M. Song, and Y. Takeuchi, Global stability of an SIR epidemic model with time delay, Appl. Math. Lett. 17 (2004), no. 10, 1141–1145 https://doi.org/10.1016/j.aml.2003.11.005
  20. A. F. Nindjin, M. A. Aziz-Alaoui, and M. Cadivel, Analysis of a predator-prey model with modified Leslie-Gower and Holling-type II schemes with time delay, Nonlinear Anal. Real World Appl. 7 (2006), no. 5, 1104–1118 https://doi.org/10.1016/j.nonrwa.2005.10.003
  21. M. Song, W. Ma, and Yasuhiro Takeuchi, Permanence of a delayed SIR epidemic model with density dependent birth rate, J. Comput. Appl. Math. 201 (2007), no. 2, 389–394 https://doi.org/10.1016/j.cam.2005.12.039
  22. X. Y. Song and Y. F. Li, Dynamic behaviors of the periodic predator-prey model with modified Leslie-Gower Holling-type II schemes and impulsive effect, Nonlinear Anal. Real World Appl. 9 (2008), no. 1, 64–79 https://doi.org/10.1016/j.nonrwa.2006.09.004
  23. Y. L. Song, M. A. Han, and J. J. Wei, Stability and Hopf bifurcation analysis on a simplified BAM neural network with delays, Phys. D 200 (2005), no. 3-4, 185–204 https://doi.org/10.1016/j.physd.2004.10.010
  24. H. R. Thieme, Mathematics in Population Biology, Princeton Series in Theoretical and Computational Biology. Princeton University Press, Princeton, NJ, 2003
  25. W. Wang and Z. Ma, Global dynamics of an epidemic model with time delay, Nonlinear Anal. Real World Appl. 3 (2002), no. 3, 365–373 https://doi.org/10.1016/S1468-1218(01)00035-9
  26. Y. Xiao and L. Chen, Modeling and analysis of a predator-prey model with disease in the prey, Math. Biosci. 171 (2001), no. 1, 59–82 https://doi.org/10.1016/S0025-5564(01)00049-9
  27. X. Yang, L. S. Chen, and J. F. Chen, Permanence and positive periodic solution for the single-species nonautonomous delay diffusive models, Comput. Math. Appl. 32 (1996), no. 4, 109–116 https://doi.org/10.1016/0898-1221(96)00129-0

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